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Group analysis of general Burgers–Korteweg–de Vries

equations

Stanislav Opanasenko†‡, Alexander Bihlo† and Roman O. Popovych‡§,

† Department of Mathematics and Statistics, Memorial University of Newfoundland,St. John’s (NL) A1C 5S7, Canada

‡ Institute of Mathematics of NAS of Ukraine, 3 Tereshchenkivska Str., 01004 Kyiv, Ukraine

§ Wolfgang Pauli Institut, Oskar-Morgenstern-Platz 1, 1090 Wien, AustriaMathematical Institute, Silesian University in Opava, Na Rybńıčku 1, 746 01 Opava, Czechia

E-mail: sopanasenko@mun.ca, abihlo@mun.ca, rop@imath.kiev.ua

The complete group classification problem for the class of (1+1)-dimensional rth order gen-eral variable-coefficient Burgers–Korteweg–de Vries equations is solved for arbitrary valuesof r greater than or equal to two. We find the equivalence groupoids of this class and itsvarious subclasses obtained by gauging equation coefficients with equivalence transforma-tions. Showing that this class and certain gauged subclasses are normalized in the usualsense, we reduce the complete group classification problem for the entire class to that forthe selected maximally gauged subclass, and it is the latter problem that is solved efficientlyusing the algebraic method of group classification. Similar studies are carried out for the twosubclasses of equations with coefficients depending at most on the time or space variable,respectively. Applying an original technique, we classify Lie reductions of equations from theclass under consideration with respect to its equivalence group. Studying alternative gaugesfor equation coefficients with equivalence transformations allows us not only to justify thechoice of the most appropriate gauge for the group classification but also to construct for thefirst time classes of differential equations with nontrivial generalized equivalence group suchthat equivalence-transformation components corresponding to equation variables locally de-pend on nonconstant arbitrary elements of the class. For the subclass of equations withcoefficients depending at most on the time variable, which is normalized in the extendedgeneralized sense, we explicitly construct its extended generalized equivalence group in arigorous way. The new notion of effective generalized equivalence group is introduced.

1 Introduction

The development of new powerful tools and techniques of group analysis of differential equationsduring the last decade has essentially extended the range of effectively solvable problems of thisbranch of mathematics. In particular, it became possible to study admissible transformationsand Lie symmetries for systems of differential equations from complex classes parameterized bya few functions of several arguments.

A number of evolution equations that are important in mathematical physics are of thegeneral form

ut + C(t, x)uux =

r∑

k=0

Ak(t, x)uk +B(t, x). (1)

Here and in the following the integer parameter r is fixed, and r > 2. We require the conditionCAr 6= 0 guaranteeing that equations from the class (1) are nonlinear and of genuine order r.Throughout the paper we use the standard index derivative notation ut = ∂u/∂t, uk = ∂

ku/∂xk,and also u0 = u, ux = u1, uxx = u2 and uxxx = u3.

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http://arxiv.org/abs/1703.06932v2

The class (1) contains a number of the prominent classical models of fluid mechanics, including

ut + uux = a2uxx Burgers equation,

ut + uux = a3uxxx Korteweg–de Vries (KdV) equation,

ut + uux = a4u4 + a3u3 + a2u2 Kuramoto–Sivashinsky equation,

ut + uux = a5u5 + a3u3 Kawahara equation,

ut + uux = arur generalized Burgers–KdV equation,

where a’s are constants, and the coefficient of the highest-order derivative is nonzero.

Due to the importance of equations from the class (1), there exist already a lot of papers inwhich particular equations of the form (1) have been considered in light of their symmetries,integrability, exact solutions, etc. Here we review only papers on admissible transformationsand group classification of classes related to the class (1).

Particular subclasses of the class (1) with small values of r were the subject of a number ofpapers published over the past twenty-five years. Thus, the equivalence groupoids of the class ofvariable-coefficient Burgers equations of the form ut+uux+f(t, x)uxx = 0 with f 6= 0 and of itssubclass of equations with f = f(t) were constructed by Kingston and Sophocleous in [14] as setsof point transformations in pairs of equations. In fact, it was implicitly proved there that theseclasses are normalized in the usual sense. Later such transformations were called allowed [10, 47]or form-preserving [15], and they had preceded the notion of admissible transformations, whichis of central importance in group analysis of classes of differential equations. Solving of thegroup classification problem for the above subclass with f = f(t) was initiated in [7, 46] andcompleted in [33, 41]. Most recently, an extended symmetry analysis of the class with f =f(t, x) considered in [14] was comprehensively carried out in [32]. The partial preliminary groupclassification problem for the related class of Burgers equations with sources, which are of theform ut+uux = uxx+f(t, x, u), with respect to the maximal Lie symmetry group of the Burgersequation was considered in [23].

In [10, 47] allowed transformations were computed for the class of variable-coefficient KdVequations of the form ut+f(t, x)uux+g(t, x)uxxx = 0 with fg 6= 0 and were then used in [10] tocarry out the group classification of this class; see [45] for a modern interpretation of these results.An attempt to reproduce these results for the class of variable-coefficient Burgers equationsof the form ut + f(t, x)uux + g(t, x)uxx = 0 with fg 6= 0 was made in [38] supposing thatadmissible transformations of this class are similar to admissible transformations of its third-order counterpart but in fact the structure of the corresponding equivalence groupoid is totallydifferent from that in [10]. Transformational properties of the class of variable-coefficient KdVequations of the form

ut + f(t)uux + g(t)uxxx + (q(t)x+ p(t))ux + h(t)u+ k(t)x+ l(t) = 0, fg 6= 0,

were studied in [37]; see also [39]. This class coincides with the class K1 (with the particular valuer = 3) arising in Section 9 of the present paper. The class Kr=31 was proved to be normalized inthe usual sense and mapped by a family of its equivalence transformations, e.g., to its subclassof equations of the form ut+uux+g(t)uxxx = 0 with g 6= 0, which is also normalized in the usualsense, and solving the group classification problem in the class Kr=31 is equivalent to that in thesubclass; cf. Proposition 7 and Section 9. Variable-coefficient generalizations of the Kawaharaequation were studied in a similar way in [17, 18]. The group classification of Galilei-invariantequations of the form ut + uux = F (ur) was carried out in [5, 8].

Unfortunately, the results obtained in many papers were incomplete or even faulty; for apartial listing of papers with such results on variable-coefficient Korteweg–de Vries equations,we refer to [37]. It is thus appropriate to solve the group classification problem for the generalclass (1) systematically for the first time.

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More general classes of evolutions equations, which include the class (1), were also consideredin the literature. Contact symmetries of (1+1)-dimensional evolution equations were studied byMagadeev [24]. More specifically, Magadeev proved that a (1+1)-dimensional evolution equationadmits an infinite-dimensional Lie algebra of contact symmetries if and only if it is linearizableby a contact transformation. He also classified, up to contact equivalence transformations ofevolution equations, realizations of finite-dimensional Lie algebras by contact vector fields withthe independent variables (t, x) and the dependent variable u each of which is the maximalcontact invariance algebras of a non-linearizable evolution equation. At the same time, evo-lution equations admitting these realizations as maximal contact invariance algebra were notconstructed. Moreover, singling out the classification of contact symmetries of equations froma particular subclass of the entire class of (1+1)-dimensional evolution equations is in general amore challenging problem than the direct classification of contact symmetries for equations fromthe subclass. A claim similar to the above one on results of [24] is also true for classifications oflow-order evolution equations (with r = 2, 3, 4) considered, e.g., in [1, 11, 12, 49].

In the present paper we solve the complete group classification problem of the class (1) forany fixed value of r > 2. An even more important result of the paper is the construction,in the course of the study of admissible transformations within subclasses of the class (1), ofseveral examples of classes with specific properties, the existence of which was in doubt for along time. These examples give us unexpected insights into the general theory of equivalencetransformations within classes of differential equations. In particular, they justify introducingthe notion of effective generalized equivalence group.

We also point out that the class of linear (in general, variable-coefficient) evolution equationsis obtained from (1) by setting C = 0. This class has transformational properties different fromthose of the class (1), and equations with C = 0 and C 6= 0 are not related to each otherby point or contact transformations. Moreover, the class of linear equations was completelyclassified in [21] and [30, pp. 114–118] for r = 2 and in [3] for r > 2. (The last paper enhancedand extended results of [11, Section III] on r = 3 and of [13] on r = 4.) This is why it isappropriate to exclude linear equations from the present consideration.

The further structure of this paper is the following. In Section 2 we present sufficient back-ground information on point transformations in classes of differential equations and on the alge-braic method of group classification. Since the class (1) is normalized in the usual sense, particu-lar emphasis is given to describing the concept of normalization of a class of differential equations.The notion of effective generalized equivalence group is introduced there for the first time. Usingknown results on admissible transformations of the entire class of rth order evolution equationsand its wide subclasses, in Section 3 we compute the equivalence groupoids for the class (1) andits two subclasses singled out by the arbitrary-element gauges C = 1 and (C,A1) = (1, 0), whichare realized by families of equivalence transformations. Due to each of the above classes beingnormalized in the usual sense, its equivalence groupoid coincides with the subgroupoid inducedby the usual equivalence group of this class. The solution of the group classification problem forthe class (1) is shown to reduce to that for its gauged subclass associated with the constraint(C,A1) = (1, 0) and referred to in the paper as the class (5). Moreover, this subclass turns outto be the most convenient for carrying out the group classification since it is normalized in theusual sense and maximally gauged. The consideration of alternative arbitrary-element gauges inSection 4 additionally justifies the selection of the subclass with (C,A1) = (1, 0) for group clas-sification. This also allows us to construct for the first time examples of classes with nontrivialgeneralized equivalence groups, where transformation components for variables locally depend onnonconstant arbitrary elements. The determining equations for Lie symmetries of equations fromthe class (1) are derived and preliminarily studied in Section 5. These results are used in Section 6for analyzing properties of appropriate (for group classification) subalgebras of the projection ofthe equivalence algebra of the class (5) to the space of equation variables. The group classificationof the gauged subclass (5) and thus the entire class (1) is completed in Section 7 via classifying

3

the appropriate subalgebras and finding the corresponding values of the arbitrary-element tuple.In Section 8, we discuss the optimality of chosen inequivalent representatives for cases of Liesymmetry extensions. The subclass K3 of equations from the class (1) with coefficients depend-ing at most on t is the object of the study in Sections 9. We exhaustively describe the equivalencegroupoid of this subclass, gauge its arbitrary elements by equivalence transformations and singleout a complete list of inequivalent Lie symmetry extensions within this subclass from that forthe class (1). In fact, we begin with the wider subclass K1, where A

1 and B may affinely dependon x, since this subclass is normalized in the usual sense with respect to its nice usual equivalencegroup whereas the subclass K2, where only B may affinely depend on x, and the subclass K3give new nontrivial examples of classes normalized only in the generalized sense and only in theextended generalized sense, respectively. Similar results for the subclass F1 of equations from theclass (1) with coefficients depending at most on x are obtained in Section 10. Since this subclassis not normalized in any appropriate sense, in order to describe its equivalence groupoid G∼F1 , it isnecessary to solve a complicated classification problem for values of the arbitrary-element tuplethat admit nontrivial admissible transformations, which are not generated by transformationsfrom the corresponding usual equivalence group. It turns out that the equivalence groupoid G∼F1has an interesting structure related to Lie symmetry extensions in the subclass F1 although thesubclass F1 is far from even being semi-normalized. Lie reductions of equations from the class (1)are classified in Section 11 with respect to the usual equivalence group of this class using the factthat it is normalized in the usual sense. Other possibilities for finding exact solutions of theseequations are also discussed. The conclusions of the paper are presented in the final Section 12.

2 Algebraic method of group classification

Basic notions and results underlying the algebraic method of group classification in its modernadvanced form as will be used below were extensively discussed in [2, 3, 19, 34, 35, 36], to whichwe refer for further details. Examples of applying various versions of the algebraic method toparticular group classification problems can be found in [1, 9, 10, 11, 13, 24, 27, 49]. In thissection we not only review the needed part of the known theory on symmetry analysis in classesof differential equations but also present some new notions and results of this field.

Let Lθ denote a system of differential equations of the form L(x, u(r), θ(x, u(r))) = 0. Here,

x = (x1, . . . , xn) are the n independent variables, u = (u1, . . . , um) are the m dependent vari-

ables, and L is a tuple of differential functions in u. We use the standard short-hand nota-tion u(r) to denote the tuple of derivatives of u with respect to x up to order r, which alsoincludes the u’s as the derivatives of order zero. The system Lθ is parameterized by thetuple of functions θ = (θ1(x, u(r)), . . . , θk(x, u(r))), called the arbitrary elements, which runsthrough the solution set S of an auxiliary system of differential equations and inequalities in θ,S(x, u(r), θ(q)(x, u(r))) = 0 and, e.g., Σ(x, u(r), θ(q)(x, u(r))) 6= 0. Here, the notation θ(q) encom-passes the partial derivatives of the arbitrary elements θ up to order q with respect to both xand u(r). Thus, the class of (systems of) differential equations L|S is the parameterized familyof systems Lθ’s, such that θ lies in S.

For the specific class of general Burgers–KdV equations (1) considered below, we have n = 2,m = 1, and, in the more traditional notation, x1 = t and x2 = x. The tuple of arbitrary elementsis θ = (A0, . . . , Ar, B,C), which runs through the solution set of the auxiliary system

Akuα = 0, k = 0, . . . , r, Buα = 0, Cuα = 0, |α| 6 r, CAr 6= 0,

where α = (α1, α2) is a multi-index, α1, α2 ∈ N ∪ {0}, |α| = α1 + α2, and uα = ∂|α|u/∂tα1∂xα2 .

Satisfying the auxiliary differential equations is equivalent to the fact that the arbitrary elementsdo not depend on derivatives of u. The inequality ArC 6= 0 ensures that equations from theclass (1) are both nonlinear and of order r.

4

Group classification of differential equations is based on studying how systems from a givenclass are mapped to each other. This study is formalized in the notion of admissible transforma-tions, which constitute the equivalence groupoid of the class L|S . An admissible transformationis a triple (θ, θ̃, ϕ), where θ, θ̃ ∈ S are arbitrary-element tuples associated with systems Lθand Lθ̃ from the class LS that are similar to each other, and ϕ is a point transformation in thespace of (x, u) that maps Lθ to Lθ̃.

A related notion of relevance in the group classification of differential equations is that ofequivalence transformations. Usual equivalence transformations are point transformations in thejoint space of independent variables, derivatives of u up to order r and arbitrary elements thatare projectable to the space of (x, u(r

′)) for each r′ = 0, . . . , r, respect the contact structureof the rth order jet space coordinatized by the r-jets (x, u(r)) and map every system from theclass L|S to a system from the same class. The Lie (pseudo)group constituted by the equivalencetransformations of L|S is called the usual equivalence group of this class and denoted by G

∼. Ifthe arbitrary elements depend at most on derivatives of u up to order r̂ < r, then one can assumethat equivalence transformations act in the space of (x, u(r̂), θ) instead of the space of (x, u(r), θ).

The usual equivalence group G∼ gives rise to a subgroupoid of the equivalence groupoid G∼

since each equivalence transformation T ∈ G∼ generates a family of admissible transformationsparameterized by θ,

G∼ ∋ T →{

(θ,T θ, π∗T ) | θ ∈ S}

⊂ G∼.

Here π denotes the projection of the space of (x, u(r), θ) to the space of equation variables only,π(x, u(r), θ) = (x, u). The pushforward π∗T of T by π is then just the restriction of T to thespace of (x, u).

The usual equivalence algebra g∼ of the class L|S is an algebra associated with the usualequivalence group G∼ and constituted by the infinitesimal generators of one-parameter groupsof equivalence transformations. These infinitesimal generators are vector fields in the jointspace of (x, u(r), θ) that are projectable to (x, u(r

′)) for each r′ = 0, . . . , r. Since equivalencetransformations respect the contact structure on the rth order jet space, the vector fields from g∼

inherit this compatibility, meaning their projections to the space of (x, u(r)) coincide with therth order prolongations of the associated projections to the space of (x, u).

In the case when the arbitrary elements θ’s are functions of (x, u) only, we can assumethat equivalence transformations of the class L|S are point transformations of (x, u, θ) mappingevery system from the class L|S to a system from the same class. The projectability property forequivalence transformations is neglected here. Then these equivalence transformations constitutea Lie (pseudo)group Ḡ∼ called the generalized equivalence group of the class L|S . See the firstdiscussion of this notion in [25, 26] with no relevant examples and the further developmentin [34, 36]. Often the generalized equivalence group coincides with the usual one; this situation isconsidered as trivial. Similar to usual equivalence transformations, each element of Ḡ∼ generatesa family of admissible transformations parameterized by θ,

Ḡ∼ ∋ T →{

(θ′,T θ′, π∗(T |θ=θ′(x,u))) | θ′ ∈ S

}

⊂ G∼,

and thus the generalized equivalence group Ḡ∼ also generates a subgroupoid H̄ of the equivalencegroupoid G∼.

Definition 1. We call any minimal subgroup of Ḡ∼ that generates the same subgroupoid of G∼

as the entire group Ḡ∼ does an effective generalized equivalence group of the class L|S .

The uniqueness of an effective generalized equivalence group is obvious if the entire group Ḡ∼

is effective itself; cf. Remark 11 below. At the same time, there exist classes of differential equa-tions, where effective generalized equivalence groups are proper subgroups of the corresponding

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generalized equivalence groups that are even not normal. Hence each of these effective gener-alized equivalence groups is not unique since it differs from some of subgroups non-identicallysimilar to it, and all of these subgroups are also effective generalized equivalence groups of thesame class. See the discussion of particular examples in Remark 28 below.

Suppose that the class L|S possesses parameterized non-identity usual equivalence transfor-mations and some of its arbitrary elements are constants. Then this class necessarily admitspurely generalized equivalence transformations. Indeed, we can set all parameters of elementsfrom the usual equivalence group G∼ depending on constant arbitrary elements, which givesgeneralized equivalence transformations. The set Ḡ∼0 of such transformations is a subgroup ofthe generalized equivalence group Ḡ∼. If Ḡ∼0 = Ḡ

∼, the usual equivalence group G∼ is aneffective generalized equivalence group of the class L|S .

The generalized equivalence algebra ḡ∼ and an effective generalized equivalence algebra of theclass L|S are the algebras associated with the generalized equivalence group Ḡ

∼ and with aneffective generalized equivalence group of this class and are constituted by the infinitesimal gen-erators of one-parameter subgroups of these groups, respectively. These infinitesimal generatorsare vector fields in the joint space of (x, u, θ).

The property for equivalence transformations to be point transformations with respect toarbitrary elements can also be weakened. We formally extend the arbitrary-element tuple θof the class L|S with virtual arbitrary elements that are related to initial arbitrary elementsby differential equations and thus expressed via initial arbitrary elements in a nonlocal way.Denote the reparameterized class by L̂|S . Suppose that the usual (resp. generalized or effectivegeneralized) equivalence group Ĝ∼ of L̂|S induces the maximal subgroupoid of the equivalencegroupoid G∼ among the classes obtained from L|S by similar reparameterizations, and theextension of the arbitrary-element tuple θ for L̂|S is minimal among the reparameterized classesgiving the same subgroupoid of G∼ as L̂|S . Then we call the group Ĝ

∼ an extended equivalencegroup (resp. an extended generalized equivalence group) of the class L|S .

A point symmetry transformation of a system Lθ is a point transformation in the spaceof (x, u) that preserves the solution set of Lθ. Each point symmetry transformation ϕ of Lθgives rise to the single admissible transformation (θ, θ, ϕ) of the class L|S . The point symmetrytransformations of the system Lθ constitute the maximal point symmetry group Gθ of thissystem. The common part G∩ of all Gθ is called the kernel of maximal point symmetry groups ofsystems from the class L|S , G

∩ :=⋂

θ∈S Gθ. The infinitesimal counterparts of the maximal pointsymmetry group Gθ and the kernel G

∩, which are called the maximal Lie invariance algebra gθof Lθ and the kernel Lie invariance algebra g

∩ of systems from L|S , consist of the vector fieldsin the space of (x, u) generating one-parameter subgroups of Gθ and G

∩, respectively.

Group analysis of differential equations becomes much simpler when working with infinitesi-mal counterparts of objects consisting of point transformations. Thus, the problem on Lie (i.e.,continuous point) symmetries of a system Lθ reduces to constructing the maximal Lie invariancealgebra gθ and, therefore, is linear in contrast to the similar problem on general point symme-tries. Under certain quite natural conditions on the system Lθ, the infinitesimal invariancecriterion states that a vector field Q in the space of (x, u) belongs to the maximal Lie invariancealgebra gθ if and only if the condition Q

(r)L(x, u(r), θ(q)(x, u(r))) = 0 is identically satisfied onthe manifold Lrθ defined by the system Lθ and its differential consequences in the jet space J

(r).Here Q(r) is the standard rth order prolongation of the vector field Q, see [29, 30] and Section 5.

The group classification problem for the class L|S is to list all G∼-inequivalent values for θ ∈ S

such that the associated systems, Lθ, admit maximal Lie invariance algebras, gθ, that are widerthan the kernel Lie invariance algebra g∩. Further taking into account additional point equiva-lences between obtained cases, provided such additional equivalences exist, one solves the groupclassification problem up to G∼-equivalence. Restricting the consideration to Lie symmetries isimportant for the group classification problem to be well-posed within the framework of classesof differential equations.

6

When applied to systems from a class L|S , the infinitesimal invariance criterion yields, aftersplitting with respect to the parametric derivatives of u, a system of the determining equations forthe components of Lie symmetry generators of these systems, which is in general parameterizedby the arbitrary-element tuple θ. It is quite common that there is a subsystem of the determiningequations that does not involve the tuple of arbitrary elements θ and hence may be integratedin a regular way. The remaining part of the determining equations that explicitly involve thearbitrary elements is referred to as the system of classifying equations. In this setting, the groupclassification problem reduces to the exhaustive investigation of the classifying equations. Thedirect integration of the classifying equations up to G∼-equivalence is usually only possible forclasses of the simplest structure, e.g., classes involving only constants or functions of a singleargument as arbitrary elements, see, e.g., examples in [30]. Since most classes of interest inapplications are of more complicated structure, various methods have to be used, which at leastenhance the direct method [28, 42, 43].

The most advanced classification techniques rest on the study of algebras of vector fieldsassociated with systems from the class L|S under consideration and constitute, in total, the al-gebraic method of group classification. For this method to be really effective, the class L|S has topossess certain properties, which are conveniently formulated in terms of various notions of nor-malization. The class of differential equations L|S is normalized in the usual (resp. generalized,extended, extended generalized) sense if the subgroupoid induced by its usual (resp. general-ized, extended, extended generalized) equivalence group coincides with the entire equivalencegroupoid G∼ of L|S .

The normalization of L|S in the usual sense is equivalent to the following conditions. Thetransformational part ϕ of each admissible transformation (θ′, θ′′, ϕ) ∈ G∼ does not depend onthe fixed initial value θ′ of the arbitrary-element tuple θ and, therefore, is appropriate for anyinitial value of θ. Moreover, the prolongation of ϕ to the space of (x, u(r)) and the furtherextension to the arbitrary elements according to the relation between θ′ and θ′′ gives a pointtransformation in the joint space of (x, u(r), θ). Then Gθ 6 π∗G

∼ and gθ ⊆ π∗g∼ for any θ ∈ S,

and hence the group classification of the class L|S reduces to the classification of certain G∼-

inequivalent subalgebras of the equivalence algebra g∼ or, equivalently, to the classification ofcertain π∗G

∼-inequivalent subalgebras of the projection π∗g∼.

If the class L|S is normalized in the generalized sense, the expression for transformationalparts of admissible transformations may involve arbitrary elements but only in a quite specificway. The equivalence groupoid is partitioned into families of admissible transformations pa-rameterized by the source arbitrary-element tuple, and the transformational parts of admissibletransformations from each of these families jointly give, after the extension to the arbitraryelements according to the relation between the corresponding source and target arbitrary ele-ments, a point transformation in the joint space of (x, u, θ). Then Gθ′ 6 π∗(G

∼|θ=θ′(x,u)) andgθ′ ⊆ π∗(g

∼|θ=θ′(x,u)) for any θ′ ∈ S.

The class L|S is called semi-normalized in the usual sense if for any (θ′, θ′′, ϕ) ∈ G∼ there

exist T ∈ G∼, ϕ′ ∈ Gθ′ and ϕ′′ ∈ Gθ′′ such that θ

′′ = T θ′ and ϕ = ϕ′′(π∗T )ϕ′. One of the

transformations ϕ′ and ϕ′′ can always be assumed to coincide with the identity transformation.Semi-normalization in the generalized sense is defined in a similar way. Roughly speaking, a classis semi-normalized in a certain sense if its equivalence groupoid is generated by its relevant equiv-alence group jointly with point symmetry groups of systems from this class. Each normalizedclass is semi-normalized in the same sense, and the converse is not in general true. There are alsomore sophisticated notions, uniform semi-normalization and weak uniform semi-normalization,which mediate the notion of normalization and semi-normalization [19, 20].

To establish the normalization properties of the class L|S one should compute its equivalencegroupoid G∼, which is realized using the direct method. Here one fixes two arbitrary systemsfrom the class, Lθ : L(x, u

(r), θ(x, u(r))) = 0 and Lθ̃ : L(x̃, ũ(r), θ̃(x̃, ũ(r))) = 0, and aims to find

the (nondegenerate) point transformations, ϕ: x̃i = Xi(x, u), ũa = Ua(x, u), i = 1, . . . , n,

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a = 1, . . . ,m, connecting them. For this, one changes the variables in the system Lθ̃ by expressing

the derivatives ũ(r) in terms of u(r) and derivatives of the functions Xi and Ua as well asby substituting Xi and Ua for x̃i and ũ

a, respectively. The requirement that the resultingtransformed system has to be satisfied identically for solutions of Lθ leads to the system ofdetermining equations for the components of the transformation ϕ.

In the case of a single dependent variable (m = 1), all the above notions involving pointtransformations can be directly extended to contact transformations.

3 Equivalence groupoid

We now compute the equivalence groupoid and equivalence group of the class (1) using thedirect method. Equivalence transformations will be used to find an appropriate gauged subclassof (1) that is suitable for carrying out the complete group classification. The presentationclosely follows [3]. In particular, it is convenient to start with the wide superclass of general(1+1)-dimensional rth order (r > 2) evolution equations of the form

ut = H(t, x, u0, . . . , ur), Hur 6= 0, (2)

and sequentially narrowing it until the class (1) and its gauged subclasses are reached. Theadvantage of this method is that one can infer the normalization properties of the class (1) bykeeping track of the normalization properties of the class (2) and its relevant subclasses. Thisnot only gives restrictions on the transformational part of admissible transformations in theclass (2) and its subclasses, but also leads to a more and more constrained relation between theinitial and target arbitrary elements until this relation is sufficiently specified.

It was established in [24] that a contact transformation of the independent variables (t, x)and the dependent variable u connects two fixed equations from the class (2) if and only if itscomponents are of the form t̃ = T (t), x̃ = X(t, x, u, ux) and ũ = U(t, x, u, ux) provided that theusual nondegeneracy assumption and contact condition hold,

Tt 6= 0, rank

(

Xx Xu XuxUx Uu Uux

)

= 2 and (Ux + Uuux)Xux = (Xx +Xuux)Uux .

The prolongation of the transformation to the first derivatives is given by

ũx̃ = V, ũt̃ =Uu −XuV

Ttut +

Ut −XtV

Tt, where V =

Ux + UuuxXx +Xuux

or V =UuxXux

if Xx + Xuux 6= 0 or Xux 6= 0, respectively. Such transformations prolonged to the arbitraryelement H according to

H̃ =Uu −XuV

TtH +

Ut −XtV

Tt

constitute the contact usual equivalence group of the class (2) and thus this class is normal-ized in the usual sense with respect to contact transformations. It is also normalized in theusual sense with respect to point transformations. The point equivalence groupoid and thepoint usual equivalence group are singled out from their contact counterparts by the conditionXux = Uux = 0.

Consider the subclass E of the class (2) singled out by the constraints

Huruk = 0, k = 1, . . . , r, Hur−1ul = 0, l = 1, . . . , r − 1,

cf. [44]. Due to the constraints Huruk = 0, k = 2, . . . , r, it follows that contact admissibletransformations in the subclass E are induced by point admissible transformations. In other

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words, the contact equivalence groupoid of E coincides with the first prolongation of the pointequivalence groupoid of E . Then the constraints Huru1 = 0 and Hur−1ul = 0, l = 1, . . . , r − 1,successively imply Xu = 0 and Uuu = 0 for any admissible transformation in E , i.e., its trans-formational part is of the form

t̃ = T (t), x̃ = X(t, x), ũ = U1(t, x)u + U0(t, x) with TtXxU1 6= 0. (3)

The prolongations of these transformations to the arbitrary element H constitute the usualequivalence group of class E . Therefore, the class E is normalized in the usual sense.

To single out the class (1), we should set more constraints for H. The complete system ofthese constraints is given by

Hukul = 0, 1 6 k 6 l 6 r, (k, l) 6= (0, 1), Hur 6= 0, Hu0u1 6= 0.

Then we should also reparameterize the obtained subclass, assuming θ = (A0, . . . , Ar, B,C)as the tuple of arbitrary elements instead of H. Using the direct method for computing theequivalence groupoid of the class (1), we first fix two arbitrary equations Lθ and Lθ̃ from theclass (1) and require that they are connected through a point transformation ϕ of the form (3).This particular form can be posed for admissible transformations since the class (1) is a subclassof the normalized class E . It is thus necessary to re-express the jet variables (t̃, x̃, ũ(r)) in termsof (t, x, u(r)). In view of (3), the expressions for the transformed total derivative operators are

Dt̃ =1

Tt

(

Dt −XtXx

Dx

)

, Dx̃ =1

XxDx.

Substituting the expressions for the transformed values into Lθ̃ yields an intermediate equa-

tion L̃. Because the equations Lθ and Lθ̃ are by assumption connected by ϕ, the equation L̃has to be satisfied by all solutions of Lθ. We assume ut as the leading derivative in Lθ andsubstitute the expression for ut obtained from Lθ into L̃. This leads to an identity, which can besplit with respect to the parametric derivatives u0, . . . , ur. The condition that the coefficient ofu2 in L̃ has to be zero requires U1x = 0. Collecting the other coefficients of powers of parametricderivatives, we derive the formulas pointwise relating θ and θ̃ with no constraints for T , X, U1

and U0. These formulas are quite cumbersome (although obtainable using the Faà di Bruno’sformula). In addition, they are not needed at the present stage since we can first fix a suitablegauged subclass of the class (1). To do this, we only need the transformation component for thearbitrary element C = C(t, x), which is readily obtained without using Faà di Bruno’s formula,

C̃ =XxTtU1

C.

Proposition 2. The class (1) is normalized in the usual sense. Its usual equivalence group G∼(1)consists of the transformations in the joint space of (t, x, u, θ) whose (t, x, u)-components are ofthe form

t̃ = T (t), x̃ = X(t, x), ũ = U1(t)u+ U0(t, x),

where T = T (t), X = X(t, x), U1 = U1(t) and U0 = U0(t, x) are arbitrary smooth functions oftheir arguments such that TtXxU

1 6= 0.

We can now use the family of equivalence transformations parameterized by the arbitraryelement C, where

t̃ = t, x̃ =

∫ x

x0

dy

C(t, y), ũ = u,

to map the class (1) onto its subclass singled out by the constraint C = 1. Under this gaugingwe derive that Xx = TtU

1 and thus Xxx = 0, i.e., X = X1(t)x +X0(t) and U1 = X1/Tt. The

gauged subclass is still normalized in the usual sense. Moreover, due to the most prominentequations from the class (1) being of the form with C = 1, this gauge is quite natural.

9

Remark 3. Given a class of differential equations, if a subclass is singled from it by constraintswith explicit expressions for some arbitrary elements, we reparameterize this subclass using thereduced tuple of arbitrary elements, which is obtained from the complete tuple by excluding theconstrained arbitrary elements. For example, under the gauge C = 1 we can assume that thetuple of arbitrary elements for the corresponding subclass is (A0, . . . , Ar, B).

With the restrictions on T , X and U obtained thus far, we now complete the procedure forfinding the equivalence groupoid of the subclass associated with gauge C = 1, which consists ofequations of the form

ut + uux =

r∑

k=0

Ak(t, x)uk +B(t, x). (4)

Transforming the equations from the gauged subclass, we find

1

Tt

(

U1ut + U1t u+ U

0t −

XtX1

(U1ux + U0x)

)

+1

X1(U1u+ U0)(U1ux + U

0x)

=

r∑

k=0

Ãk

(X1)k(U1uk + U

0k ) + B̃,

which can be written, after substituting for ut in view of the equation (4), as

r∑

k=0

Akuk +B +

(

U0

U1−XtX1

)

ux +

(

U1tU1

+U0xU1

)

u+U0tU1

−XtX1

U0xU1

+U0U0x(U1)2

=

r∑

k=0

[

Tt(X1)k

Ãkuk +TtU1

(

B̃ +Ãk

(X1)kU0k

)]

.

Splitting this equation with respect to uk yields the transformation components for the arbitraryelements. We have thus proved the following theorem.

Theorem 4. The class (4) of reduced (1+1)-dimensional general rth order Burgers–KdV equa-tions with C = 1 is normalized in the usual sense. Its usual equivalence group G∼(4) is constitutedby the transformations of the form

t̃ = T (t), x̃ = X1(t)x+X0(t), ũ =X1

Ttu+ U0(t, x),

Ãj =(X1)j

TtAj , Ã1 =

X1

TtA1 + U0 −

X1t x+X0t

Tt, Ã0 =

1

Tt

(

A0 +X1tX1

−TttTt

+TtX1

U0x

)

,

B̃ =X1

(Tt)2B +

U0tTt

+U0xX1

(

U0 −X1t x+X

0t

Tt

)

−r∑

k=0

U0k(X1)k

Ãk,

where j = 2, . . . , r, and T = T (t), X1 = X1(t), X0 = X0(t) and U0 = U0(t, x) are arbitrarysmooth functions of their arguments such that TtX

1 6= 0.

At this stage, it is convenient to introduce one more gauge. In particular, the family ofequivalence transformations parameterized by the arbitrary element A1 with T = t, X1 = 1,X0 = 0 and U0 = −A1 maps the associated gauged subclass (4) with C = 1 to the subclass of (1)with C = 1 and A1 = 0. This gauging implies that U0 = (X1t x +X

0t )/Tt. The corresponding

gauged subclass consisting of equations of the form

Lκ : ut + uux =r∑

j=2

Aj(t, x)uj +A0(t, x)u+B(t, x), (5)

10

where Ar 6= 0 and κ = (A0, A2, . . . , Ar, B) is the reduced arbitrary-element tuple, is still normal-ized in the usual sense. This is the gauged subclass that is appropriate for solving the completegroup classification problem for the class (1). This leads to the following theorem.

Theorem 5. The class (5) of reduced (1+1)-dimensional general rth order Burgers–KdV equa-tions, which is singled out from the class (1) by the gauge (C,A1) = (1, 0), is normalized in theusual sense. Its usual equivalence group G∼ consists of the transformations of the form

t̃ = T (t), x̃ = X1(t)x+X0(t), ũ =X1

Ttu+

X1tTtx+

X0tTt, (6a)

Ãj =(X1)j

TtAj , Ã0 =

1

Tt

(

A0 + 2X1tX1

−TttTt

)

, (6b)

B̃ =X1

(Tt)2B +

1

Tt

(

X1tTt

)

t

x+1

Tt

(

X0tTt

)

t

−

(

X1tTtx+

X0tTt

)

Ã0, (6c)

where j = 2, . . . , r, and T = T (t), X1 = X1(t) and X0 = X0(t) are arbitrary smooth functionsof their arguments with TtX

1 6= 0.

Corollary 6. The equivalence algebra of the class (5) of (1+1)-dimensional general rth orderBurgers–KdV equations is given by g∼ = 〈D̂(τ), Ŝ(ζ), P̂ (χ)〉, where τ = τ(t), ζ = ζ(t) andχ = χ(t) run through the set of smooth functions of t, with

D̂(τ) = τ∂t − τtu∂u − τt

r∑

j=2

Aj∂Aj − (τtA0 + τtt)∂A0 − 2τtB∂B,

Ŝ(ζ) = ζx∂x + (ζu+ ζtx)∂u + jζ

r∑

j=2

Aj∂Aj + 2ζt∂A0 + (ζB + ζttx− ζtxA0)∂B ,

P̂ (χ) = χ∂x + χt∂u + (χtt − χtA0)∂B .

Proof. Since we have already computed the usual equivalence group G∼, the associated equiv-alence algebra g∼ can be obtained in a straightforward deductive fashion. In particular, g∼ isspanned by vector fields representing the infinitesimal generators of one-parameter subgroups ofthe usual equivalence group G∼. Thus, successively assuming one of the parameter functions T ,X1 and X0 to depend on a continuous group parameter ε (in such a manner that the identicaltransformation corresponds to the value ε = 0), we can obtain the coefficients of the infinitesimalgenerators of the form Q̂ = τ∂t + ξ∂x + η∂u + φ

0∂A0 +∑r

j=2 φj∂Aj + ψ∂B by determining

τ =dt̃

dε

∣

∣

∣

ε=0, ξ =

dx̃

dε

∣

∣

∣

ε=0, η =

dũ

dε

∣

∣

∣

ε=0, φ0 =

dÃ0

dε

∣

∣

∣

ε=0, φj =

dÃj

dε

∣

∣

∣

ε=0, ψ =

dB̃

dε

∣

∣

∣

ε=0.

This results in the generating vector fields D̂(τ), Ŝ(ζ) and P̂ (χ), which are associated to theparameter functions T , X1 and U0, respectively.

Since the class (1) is mapped onto the class (5) by a family of equivalence transformations,and both the classes are normalized in the usual sense, the following assertion is obvious.

Proposition 7. The group classification of the class (1) reduces to that of its subclass (5). Morespecifically, any complete list of G∼-inequivalent Lie symmetry extensions in the class (5) is acomplete list of G∼(1)-inequivalent Lie symmetry extensions in the class (1).

11

4 Alternative gauges

We show that the gauge C = 1 is the best initial gauge for the class (1) and the gauge (C,A1) =(1, 0) is the best for singling out a subclass in order to carry out the group classification.

An obvious choice for an alternative gauge is Ar = 1. It was used in [3] as the basic gaugein the course of group classification of linear equations of the form (1), for which C = 0. TheAr-component of equivalence transformations in the class (1) is

Ãr =(Xx)

r

TtAr.

If Ar = 1 and Ãr = 1, then the parameters of the corresponding admissible transformations givenin Proposition 2 satisfy the constraint (Xx)

r = Tt, i.e., X = X1(t)x+X0(t), where (X1)r = Tt,

which makes the parameterization of the usual equivalence group of the corresponding subclassmore complicated than using the gauge C = 1.

Proposition 8. The subclass of the class (1) singled out by the constraint Ar = 1 is normalizedin the usual sense. Its usual equivalence group is constituted by the transformations of the form

t̃ = T (t), x̃ = X1(t)x+X0(t), ũ = U1(t)u+ U0(t, x),

Ãl =(X1)l

TtAl, Ã1 =

X1

Tt

(

A1 +U0

U1C −

X1t x+X0t

X1

)

, Ã0 =1

Tt

(

A0 +U1tU1

+U0xU1

C

)

,

B̃ =U1

TtB +

U0tTt

+U0xTt

(

U0

U1C −

X1t x+X0t

X1

)

−U0r

(X1)r−

r−1∑

k=0

U0k(X1)k

Ãk, C̃ =X1

TtU1C,

where l = 2, . . . , r − 1, and T = T (t), X0 = X0(t), U1 = U1(t) and U0 = U0(t, x) are arbitrarysmooth functions of their arguments such that TtU

1 6= 0, as well as X1 = (Tt)1/r if r is odd and

Tt > 0, X1 = ε(Tt)

1/r with ε = ±1 if r is even.

In contrast to the class (4), the additional gauge A1 = 0 slightly worsens the normalizationproperty. It leads to the appearance of the arbitrary element C in the u-component of admissibletransformations since then we have

U0 =X1t x+X

0t

X1CU1.

Denote by θ′ the arbitrary-element tuple reduced by the double gauge,

θ′ = (A0, A2, . . . , Ar−1, B,C).

Proposition 9. The equivalence groupoid of the subclass A1 of the class (1) singled out by theconstraints Ar = 1 and A1 = 0 consists of the triples (θ′, θ̃′, ϕ)’s, where the point transforma-tion ϕ is of the form

t̃ = T (t), x̃ = X1(t)x+X0(t), ũ = U1(t)u+ U0, U0 :=X1t x+X

0t

X1CU1, (7a)

the arbitrary-element tuples θ′ and θ̃′ are related according to

Ãl =(X1)l

TtAl, Ã0 =

1

Tt

(

A0 +U1tU1

+U0xU1

C

)

, C̃ =X1

TtU1C, (7b)

B̃ =U1

TtB +

U0tTt

+U0xTt

(

U0

U1C −

X1t x+X0t

X1

)

−U0r

(X1)r−

r−1∑

l=2

U0l(X1)l

Ãl − U0Ã0, (7c)

with l = 2, . . . , r − 1, and T = T (t), X0 = X0(t) and U1 = U1(t) being arbitrary smoothfunctions of t such that TtU

1 6= 0, as well as X1 = (Tt)1/r if r is odd and Tt > 0, X

1 = ε(Tt)1/r

with ε = ±1 if r is even.

12

It is obvious that the subclass A1 is not normalized in the usual sense. Its usual equivalencegroup is constituted by the point transformations of the form (7) in the joint space of the variables(t, x, u) and the arbitrary elements θ′, where parameters satisfy more constraints, Ttt = X

0t = 0,

and thus X1t = 0 and U0 = 0.

All the components of (7) locally depend on C, and, moreover, the expressions for Ã0 and B̃involve derivatives of C with respect to t and x. This is why, to interpret (7) as generalizedequivalence transformations, we need to formally extend the arbitrary-element tuple θ′ with thederivatives of C as new arbitrary elements, Z0 := Ct and Z

k := Ck, k = 1, . . . , r, and prolongequivalence transformations to them,

Z̃0 =X1

T 2t U1Z0 +

(

X1

TtU1

)

t

C

Tt, Z̃k =

(X1)1−k

T 2t U1Zk, k = 1, . . . , r. (8)

The derivatives of U0 in the expressions for Ã0 and B̃ should be expanded and then derivativesof C should be replaced by the corresponding Z’s.

We denote by Ā1 the class of equations of the form (1) with (Ar, A1) = (1, 0) and the

extended arbitrary-element tuple θ̄′ = (A0, A2, . . . , Ar−1, B,C,Z0, . . . , Zr), where the relationsdefining Z0, . . . , Zr are assumed as additional auxiliary equations for arbitrary elements.

Theorem 10. The class Ā1 is normalized in the generalized sense. Its generalized equivalencegroup Ḡ∼

Ā1coincides with its effective generalized equivalence group and consists of the point

transformations in the joint space of the variables (t, x, u) and the arbitrary elements θ̄′ withcomponents of the form (7), (8) and the same constraints for parameters as in Proposition 9,where partial derivatives of U0 are replaced by the corresponding restricted total derivatives withD̄t = ∂t + Z

0∂C and D̄x = ∂x + Z1∂C + Z

2∂Z1 + · · ·+ Zr∂Zr−1.

Proof. The point transformations of the above form constitute a group G, which generatesthe entire equivalence groupoid of the class Ā1 and is minimal among point-transformationgroups in the joint space of (t, x, u, θ̄′) that have this generation property. Therefore, G is aneffective generalized equivalence group of the class Ā1. We are going to prove that the group Gcoincides with Ḡ∼

Ā1. Indeed, substituting every particular value of θ̄′ to any element of Ḡ∼

Ā1gives an admissible transformation of the class Ā1. This implies that elements of Ḡ

∼Ā1

are of theform (7), (8), where the parameter functions T , X0 and X1 may depend on arbitrary elements,and the partial derivatives of these functions are replaced by the corresponding total derivativesprolonged to the arbitrary elements of the class Ā1. At the same time, these parameters satisfythe condition DxT = DxX

0 = DxX1 = 0 with the prolonged total derivative operator Dx. This

condition implies via splitting with respect to unconstrained derivatives of arbitrary elementsthat the parameters T , X0 and X1 are functions of t only. Hence Ḡ∼

Ā1= G.

Therefore, the gauge (C,A1) = (1, 0) is better than the gauge (Ar, A1) = (1, 0).

Remark 11. To the best of our knowledge, Theorem 10 provides the first example for a gen-eralized equivalence group containing transformations whose components for equation variablesdepend on a nonconstant arbitrary element. This is also an example of a generalized equivalencegroup being effective itself, and thus the corresponding class of differential equations admits aunique effective generalized equivalence group.

A more complicated example of a generalized equivalence group is given by the subclass A0of the class (1) singled out by the mere constraint A1 = 0. The A1-component of equivalencetransformations of the class (1) takes the form

Ã1 =XxTtA1 +

XxTt

U0

U1C −

XtTt

−r∑

j=2

ÃjXx

(

1

Xx∂x

)j−1 1

Xx,

13

where each Ãj , j = 2, . . . , r, is a combination of Ai, i = j, . . . , r with coefficients expressed viaTt and derivatives of X with respect to x. Substituting the expression for U

0 implied by thegauge A1 = 0,

U0 =XtU

1

XxC+TtU

1

C

r∑

j=2

Ãj(

1

Xx∂x

)j−1 1

Xx,

into the general form of admissible transformations of the class (1) and neglecting the rela-tion between A1 and Ã1, we get the elements of the equivalence groupoid of the subclass A0.Therefore, this subclass is not normalized in the usual sense, and its usual equivalence groupis isomorphic to the subgroup of the group G∼(1) singled out by constraining group parameterswith Xxx = Xt = 0. Similarly to Theorem 10, we can consider the counterpart Ā0 of thesubclass A0, where the tuple of arbitrary elements (A

0, A2, . . . , Ar, B,C) is formally extendedwith the derivatives Ct, Ck, A

jt and A

jk, j = 2, . . . , r, k = 1, . . . , r. Then the expressions for

transformational parts of admissible transformations of A0 and their relations between initialand transformed arbitrary elements including the prolongation to the above derivatives givethe components of the transformations constituting a group G, which is obviously an effectivegeneralized equivalence group G of the class Ā0. Thus, the class Ā0 is also normalized in thegeneralized sense.

Note that the entire generalized equivalence group Ḡ∼Ā0

of the class Ā0 coincides with itseffective generalized equivalence group G. Indeed, in view of the description of the equiva-lence groupoid of the subclass A0, elements of Ḡ

∼Ā0

are of the form similar to that of elementsof G, where the group parameters T , X and U1 may also depend on arbitrary elements of thesubclass A0, and their partial derivatives in t and x are replaced by the corresponding totalderivatives prolonged to the arbitrary elements of the class Ā0. At the same time, the conditionDxT = DxU

1 = 0 with the prolonged total derivative operator Dx implies that the parameterfunctions T and U1 still depend at most on t. The corresponding expression for U0 involvesthe derivative DrxX, and hence the transformation component for B necessarily contains thederivative D2rx X. Splitting this component with respect to the 2rth order x-derivatives of allarbitrary elements, which are not constrained, we derive that in fact the parameter function Xalso does not depend on arbitrary elements.

5 Preliminary analysis of Lie symmetries

We compute the maximal Lie invariance group of an equation Lκ from the class (5) using theinfinitesimal method. For this, we define the generators of one-parameter point symmetry groupsof Lκ through Q = τ∂t + ξ∂x + η∂u with the components τ , ξ and η depending on (t, x, u). Theinfinitesimal invariance criterion reads

Q(r)(

ut + uux −r∑

j=2

Ajuj −A0u−B

)

= 0 for all solutions of Lκ.

The rth prolongation Q(r) of the vector field Q is given by Q(r) = Q +∑

0

The infinitesimal invariance criterion yields

η(1,0) + uη(0,1) + ηux =

r∑

j=2

[

(τAjt + ξAjx)uj +A

jη(0,j)]

+ (τA0t + ξA0x)u+A

0η

+ τBt + ξBx, wherever ut + uux =r∑

j=2

Ajuj +A0u+B.

(9)

We have shown in Section 3 that the class (5) is normalized in the usual sense. Therefore, weknow that the restrictions derived in Corollary 6 for the (t, x, u)-components of vector fields fromthe equivalence algebra g∼ also hold for the components of infinitesimal symmetry generators.It is thus true that τ = τ(t), ξ = ζ(t)x+ χ(t), η = (ζ(t)− τt(t))u+ ζt(t)x+ χt(t).

Substituting this restricted form of the coefficients of Q into the infinitesimal invariancecriterion (9), we obtain

r∑

j=2

(τAjt + (ζx+ χ)Ajx + (τt − jζ)A

j)uj + (τA0t + (ζx+ χ)A

0x + τtA

0)u

+ τBt + (ζx+ χ)Bx − (ζ − 2τt)B + (ζtx+ χt)A0 = (2ζt − τtt)u+ ζttx+ χtt.

This equation can be split with respect to u and its spatial derivatives, resulting in the system

τAjt + (ζx+ χ)Ajx + (τt − jζ)A

j = 0, j = 2, . . . , r, (10a)

τA0t + (ζx+ χ)A0x + τtA

0 = 2ζt − τtt, (10b)

τBt + (ζx+ χ)Bx − (ζ − 2τt)B + (ζtx+ χt)A0 = ζttx+ χtt. (10c)

Since all of the determining equations (10) essentially depend on the arbitrary elements κ,they constitute the system of classifying equations for Lie symmetries of equations from theclass (5). Thus, solving the group classification problem for the class (5) reduces to solving theclassifying equations (10) up to the equivalence induced by G∼. Due to the structure of thedetermining equations (10) we have proved the following proposition.

Proposition 12. The maximal Lie invariance algebra gκ of the equation Lκ from the class (5)is spanned by the vector fields of the form Q = D(τ) + S(ζ) + P (χ), where the parameterfunctions τ , ζ and χ run through the solution set of the determining equations (10), and

D(τ) = τ∂t − τtu∂u, S(ζ) = ζx∂x + (ζu+ ζtx)∂u, P (χ) = χ∂x + χt∂u.

Proposition 13. The kernel Lie invariance algebra g∩ :=⋂

κ gκ of equations from the class (5)is trivial, that is, g∩ = {0}.

Proof. To derive the kernel of maximal Lie invariance algebras one assumes the arbitrary ele-ments κ to vary. Then it is possible to split the determining equations (10) also with respect tothe arbitrary elements κ and their derivatives. This immediately yields τ = 0, ζx+ χ = 0 and,since ζ and χ are functions of t only, it follows that ζ = χ = 0.

At this stage, it is appropriate to introduce the linear span

g〈 〉 := 〈D(τ), S(ζ), P (χ)〉,

where the parameter functions τ , ζ and χ run through the set of smooth functions of t. Thenonzero commutation relations between vector fields spanning g〈 〉 are given by

[D(τ),D(τ̌ )] = D(τ τ̌t − τ̌ τt), [D(τ), S(ζ)] = S(τζt), [D(τ), P (χ)] = P (τχt),

[S(ζ), P (χ)] = −P (ζχ).

15

In view of these commutation relations, it is obvious that g〈 〉 is a Lie algebra. Moreover, it istrue that g〈 〉 = ∪κ gκ since any of the vector fields D(τ), S(ζ) and P (χ) lies in gκ for someparticular value of the arbitrary element κ.

Let us here and in the following denote by π the projection map from the joint space of(t, x, u, κ) onto the space of the variables (t, x, u) alone, i.e., π(t, x, u, κ) = (t, x, u). This mapproperly pushforwards vector fields from g∼ and transformations from G∼, and π∗g

∼ = g〈 〉.This displays that in fact the class (5) is strongly normalized in the usual sense [36]. Notethat the normalization of the class (5) in the usual sense only implies that g〈 〉 ⊆ π∗g

∼. Thespanning vector fields of g∼, D̂(τ), Ŝ(ζ) and P̂ (χ), are pushforwarded by π to the spanningvector fields D(τ), S(ζ) and P (χ) of g〈 〉, respectively. By means of the pushforward by π, theadjoint action of G∼ on g∼ induces the action of π∗G

∼ on g〈 〉 and, therefore, on the set ofsubalgebras of g〈 〉. Recall that a subalgebra s of g〈 〉 is called appropriate if there exists a κsuch that s = gκ. For any value of the arbitrary element κ and any transformation T ∈ G

∼,the pushforward by π∗T maps the maximal Lie invariance algebra gκ of the equation Lκ ontothe maximal Lie invariance algebra gT κ of the equation LT κ, and both the algebras gκ and gT κare included in g〈 〉. In other words, the action of π∗G

∼ on g〈 〉 preserves the set of appropriatesubalgebras of g〈 〉, and hence the group π∗G

∼ generates a well-defined equivalence relation onthese subalgebras. As a result, we have proved the following proposition, which is the basis forthe group classification of class (5).

Proposition 14. The complete group classification of the class (5) of gauged (1+1)-dimensionalgeneral Burgers–KdV equations of order r is obtained by classifying all appropriate subalgebrasof the Lie algebra g〈 〉 under the equivalence relation generated by the action of π∗G

∼.

To efficiently carry out the group classification using the algebraic method, it is necessary tocompute the adjoint actions of the transformations from π∗G

∼ on the vector fields Q from g〈 〉.The adjoint actions of the transformations, ϕ, from π∗G

∼ on the vector fields, Q, from g〈 〉are directly computable from the definition of the pushforward ϕ∗ of Q by ϕ,

ϕ∗Q = Q(T )∂t̃ +Q(X)∂x̃ +Q(U)∂ũ,

see, e.g., [2, 3, 6, 19]. Here, the coefficients of ϕ∗Q are given in terms of the transformed variablesobtained by substituting (t, x, u) = ϕ−1(t̃, x̃, ũ) using the inverse transformation ϕ−1 of ϕ.

In practice, this is done by considering the families of elementary transformations D(T ),S(X1) and P(X0) from π∗G

∼, which follow from (6) by restricting all except one of the parameterfunctions T , X1 and X0 to trivial values (which are t for T , one for X1 and zero for X0). Thenontrivial pushforwards of the spanning vector fields of g〈 〉 by these elementary transformationsfrom π∗G

∼ are given by

D∗(T )D(τ) = D̃(τTt), D∗(T )S(ζ) = S̃(ζ), D∗(T )P (χ) = P̃ (χ),

S∗(X1)D(τ) = D̃(τ) + S̃

(

τX1tX1

)

, S∗(X1)P (χ) = P̃ (χX1),

P∗(X0)D(τ) = D̃(τ) + P̃

(

τX0t)

, P∗(X0)S(ζ) = S̃(ζ)− P̃ (ζX0).

(11)

Here the tildes over the right-hand side operators indicate that the given vector fields are ex-pressed using the transformed variables, which also includes substituting t = T−1(t̃) for t, whereT−1 is the inverse function of T .

6 Properties of appropriate subalgebras

An important step for the group classification of class (5) is to study properties of appropriatesubalgebras of the algebra g〈 〉 to be classified. In particular, we determine the maximum dimen-sion of admitted Lie invariance algebras of equations from this class. This is formulated in thefollowing lemma.

16

Lemma 15. For any tuple of arbitrary elements κ, dim gκ 6 5.

Proof. This statement follows directly from analyzing the solution space of the linear system (10)with respect to τ , ζ and χ for a fixed tuple κ. Denote by Ωt ⊆ R and Ωx ⊆ R open intervals onthe t- and x-axes, such that the equation Lκ is defined on the domain Ωt × Ωx. Since A

r 6= 0by definition, we can resolve the equation (10a) with j = r for τt and fix x1 ∈ Ωx, yielding

τt = rζ −

(

τArtAr

+ (ζx+ χ)ArxAr

)∣

∣

∣

∣

x=x1

=: R1. (12a)

Evaluating the classifying condition (10c) at the two distinct points x2 and x3 from Ωx andvarying t, we obtain

ζttx2 − χtt = R2, ζttx3 − χtt = R

3,

where R2 and R3 follow from substituting x2 and x3 into the left hand side of (10c), respectively.Due to x2 and x3 being different points, the above system can be written as

ζtt = · · · , χtt = · · · . (12b)

If the arbitrary element κ is fixed, the system (12) can be considered as a canonical systemof linear ordinary differential equations in t for τ , ζ and χ. Its solution space is thus five-dimensional. Further conditions derived from the classifying equations (10) can only reduce thissolution space and hence it follows that dim gκ 6 5.

We introduce three integers related to the dimensions of certain subspaces of the maximalLie invariance algebra gκ of the equation Lκ,

k1 := dim(

gκ ∩ 〈P (χ)〉)

,

k2 := dim(

gκ ∩ 〈S(ζ), P (χ)〉)

− k1,

k3 := dim gκ − dim(

gκ ∩ 〈S(ζ), P (χ)〉)

= dim gκ − k1 − k2.

Although these integers depend on κ, in view of (11) it is obvious that the dimensions of thesubalgebras gκ∩〈P (χ)〉 and gκ∩〈S(ζ), P (χ)〉 as well as of the entire algebra gκ are G

∼-invariant,and thus the integers k’s are also G∼-invariant.

Lemma 16. The integers k1, k2 and k3 are G∼-invariant values, i.e., they are the same for all

G∼-equivalent equations from the class (5).

Proof. Let T ∈ G∼ transform Lκ to Lκ̃. The transformation π∗T pushforwards gκ onto gκ̃. Atthe same time, it preserves the spans 〈S(ζ), P (χ)〉 and 〈P (χ)〉. This is why dim gκ = dim gκ̃,dim gκ ∩ 〈S(ζ), P (χ)〉 = dim gκ̃ ∩ 〈S(ζ), P (χ)〉 and dim gκ ∩ 〈P (χ)〉 = dim gκ̃ ∩ 〈P (χ)〉.

We now proceed to find the upper bounds for values of these integers, which is proved in away similar to Lemma 15.

Lemma 17. For any κ, we have (k1, k2) ∈ {(0, 0), (0, 1), (2, 0)}.

Proof. For any vector field Q = S(ζ) + P (χ) from the algebra gκ, the parameter functions ζand χ satisfy the system of classifying equations (10) for the chosen tuple of arbitrary elementsκ = (A0, A2, . . . , Ar, B) and τ = 0.

In particular, if Arx 6= 0, we can solve the classifying equation (10a) with j = r with respectto χ to obtain χ = (rAr/Arx − x)ζ. After fixing a value x = x0, this equation implies that thereexists a function f = f(t) such that χ = fζ. Then the classifying equation (10b) implies that2ζt = (x+ f)A

0xζ. Again fixing x = x0, we hence derive an equation ζt = gζ with some function

17

g = g(t). Since the parameter functions ζ and χ of any vector field Q = S(ζ) + P (χ) from gκsatisfy the same system χ = fζ and ζt = gζ, we have k1 = 0 and k2 6 1 in this case.

If Arx = 0, then the classifying equation (10a) with j = r directly gives ζ = 0, i.e., k2 = 0.Suppose that k1 > 0, i.e., there exists a nonzero χ

1 such that P (χ1) ∈ gκ. It then followsfrom the system (10) that Ajx = 0, A0x = 0 and χ

1Bx + χ1tA

0 = χ1tt. Differentiating the lastequation with respect to x results in Bxx = 0. Thus, we have P (χ) ∈ gκ for any χ from thetwo-dimensional solution space of the equation χtt = A

0χt +Bxχ, which means k1 = 2.

The projection ̟ from the space of (t, x, u) on the space of t alone properly pushforwardselements of g〈 〉 according to D(τ) + S(ζ) + P (χ) 7→ τ∂t, and hence ̟∗g〈 〉 = {τ∂t}, where τruns through the set of smooth functions of t. The pushforward ̟∗G

∼ of G∼ by ̟ is also welldefined.

Lemma 18. The projection ̟∗gκ is a Lie algebra for any tuple of arbitrary elements κ, anddim̟∗gκ = k3 6 3. It is true that ̟∗gκ ∈ {0, 〈∂t〉, 〈∂t, t∂t〉, 〈∂t, t∂t, t

2∂t〉} mod ̟∗G∼.

Proof. We show that ̟∗gκ is indeed a Lie algebra. Given τi∂t ∈ ̟∗gκ, i = 1, 2, there exist

Qi ∈ gκ such that ̟∗Qi = τ i∂t. For any constants c1 and c2, we have that c1Q

1 + c2Q2 ∈ gκ

and thus c1τ1∂t + c2τ

2∂t = ̟∗(c1Q1 + c2Q

2) ∈ ̟∗gκ . This means that ̟∗gκ is indeed a linearspace. This space is closed under the Lie bracket of vector fields and, therefore, is a Lie algebra,because [τ1∂t, τ

2∂t] = (τ1τ2t − τ

2τ1t )∂t = ̟∗[Q1, Q2] ∈ ̟∗gκ.

Since the pushforward ̟∗G∼ of G∼ by the projection ̟ coincides with the (pseudo)group

of local diffeomorphisms in the space of t, Lie’s theorem can be invoked. It states that themaximum dimension of finite-dimensional Lie algebras of vector fields on the complex (resp.real) line is three. Up to local diffeomorphisms of the line, these algebras are given by {0}, 〈∂t〉,〈∂t, t∂t〉 and 〈∂t, t∂t, t

2∂t〉.

It follows that |k| := k1 + k2 + k3 = dim gκ 6 5.

7 Group classification

The main result of the paper is given by the following assertion.

Theorem 19. A complete list of G∼-inequivalent (and, therefore, G∼-inequivalent) Lie symme-try extensions in the class (5) is exhausted by the cases given in Table 1.

Proof. Lemmas 17–18 imply that any appropriate subalgebra s of g〈 〉 has a basis consisting of

1. k1 vector fields Qi = P (χi), i = 1, . . . , k1, with linearly independent χ’s,

2. k2 vector fields Qi = S(ζ i) + P (χi), i = k1 + 1, . . . , k1 + k2, with nonzero ζ’s,

3. k3 vector fields Qi = D(τ i)+S(ζ i)+P (χi), i = k1 + k2 +1, . . . , |k|, with linearly indepen-

dent τ ’s,

where (k1, k2) ∈ {(0, 0), (0, 1), (2, 0)} and k3 6 3. The proof proceeds by separately investigatingthe cases associated with the possible range of the tuple of invariant integers (k1, k2, k3). Foreach possible value of (k1, k2, k3), we start with the above form of basis vector fields Q’s of sand simplify them as much as possible using the adjoint actions of equivalence transformationspresented in (11) and linear recombination of Q’s. At the same time, we take into account thefact that s is a Lie algebra, i.e., it is closed with respect to the Lie bracket of vector fields,[Qi

′, Qi

′′] ∈ 〈Qi, i = 1, . . . , |k|〉, i′, i′′ = 1, . . . , |k|. This leads to constraints for the components

of Q’s only if k2 + k3 > 1 or k2 + k3 = 1 and k1 = 2. Substituting the components of eachsimplified Qi into the system of classifying equations (10) yields a system of equations in κ for

18

Table 1. Complete group classification of the class (1) (resp. (5)).

no. κ Basis of gκ

0 Aj = Aj(t, x), A0 = A0(t, x), B = B(t, x) —

1 Aj = Aj(x), A0 = A0(x), B = B(x) D(1)

2 Aj = ajex, A0 = a0e

x, B = be2x D(1),D(t) − P (1)

3 Aj = ajxj |x|ν , A0 = a0|x|

ν , B = bx|x|2ν D(1),D(t) − S(ν−1), ν 6= 0

4 Aj = ajxj−2, A0 = 0, B = bx

−3 D(1),D(t) + S(12),D(t2) + S(t)

5 Aj = αj(t)xj , A0 = 0, B = β(t)x S(1)

6 Aj = αj(t)xj , A0 = 1+2 ln |x|, B = β(t)x−x ln2 |x| S(et)

7 Aj = ajxj , A0 = 0, B = bx S(1),D(1)

8 Aj = ajxj , A0 = 1 + 2 ln |x|, B = bx− x ln2 |x| S(et),D(1)

9 Aj = αj(t), A0 = B = 0 P (1), P (t)

10 Aj = aj , A0 = a0, B = bx P (χ

1), P (χ2),D(1)

11 Ar = 1, Aj = 0, j 6= r, A0 = B = 0 P (1), P (t),D(1),D(t) + S(1r )

and, for r = 2, D(t2) + S(t)

Here j = 2, . . . , r, C = 1, A1 = 0, D(τ ) = τ∂t − τtu∂u, S(ζ) = ζx∂x + (ζu + ζtx)∂u and P (χ) = χ∂x + χt∂u.The parameters αj , α0 and β are arbitrary smooth functions of t with αr 6= 0. The parameters aj , a0 and b arearbitrary constants with ar 6= 0. In Case 10, the parameter functions χ1 and χ2 are linearly independent solutionsof the equation χtt−a0χt−bχ = 0, and thus there are three cases for them depending on the sign of ∆ := a20−4b,

(a) χ1 = eλ1t, χ2 = eλ2t if ∆ > 0, where λ1,2 = (a0 ±√∆)/2;

(b) χ1 = eµt, χ2 = teµt if ∆ = 0, where µ = a0/2;(c) χ1 = eµt cos νt, χ2 = eµt sin νt if ∆ < 0, where µ = a0/2 and ν =

√−∆/2.

ar = 1 mod G∼ in Cases 2, 3, 4, 7 and 10. Moreover, in Case 2 one of the constants aj ’s with j < r, a0 or b, if it

is nonzero, can be set to ±1 by shifts of x. See also Section 10 for the justification of gauging of parameters andRemark 22 for the conditions under which the presented vector fields really span the corresponding maximal Lieinvariance algebra.

which gκ ⊃ s. In total, we have |k| such systems. We unite them and simultaneously solvefor the arbitrary elements κ. The compatibility of the joint system with respect to κ mayimply additional constraints for the components of Q’s. The expression obtained for κ can besimplified by equivalence transformations whose projected adjoint actions preserve s. ExceptCases 0 and 1, the condition for κ with gκ = s is obtained by the negation of the correspondingcondition represented in Remark 22.

We have to consider the following cases:

k1 = k2 = 0. Here dim s = k3. For k3 > 1, in view of Lemma 18 we can use the simplified formof Qi, i = 1, . . . , k3 − 1, derived in the subcase with the preceding value of k3.

k3 = 0. We obtain the general Case 0, where the algebra s coincides with the kernel algebrag∩ = {0} and thus there are no constraints for κ.

k3 = 1. A basis of s consists of a single vector field Q1 with τ1 6= 0. Successively using the

adjoint actions D∗(T ), S∗(X1) and P∗(X

0) given in (11) for appropriate functionsX0, X1 and T ,this vector field can be mapped to Q1 = D(1). The system of classifying equations (10) withthe components of Q1 evidently is Ajt = A

0t = Bt = 0, which results in Case 1 of Table 1.

19

k3 = 2. Modulo π∗G∼-equivalence, basis elements of s take the form Q1 = D(1) and Q2 =

D(t) + S(ζ2) + P (χ2). The condition [Q1, Q2] = D(1) + S(ζ2t ) + P (χ2t ) ∈ 〈Q

1, Q2〉 requires thatζ2t = χ

2t = 0 and thus ζ

2, χ2 = const. Substituting the components of Q1 and then Q2 into theclassifying equations (10) and rearranging leads to the system

Ajt = 0, (ζ2x+ χ2)Ajx + (1− jζ

2)Aj = 0,

A0t = 0, (ζ2x+ χ2)A0x +A

0 = 0,

Bt = 0, (ζ2x+ χ2)Bx + (2− ζ

2)B = 0.

(13)

If ζ2 = 0, in view of the condition Ar 6= 0 the equation (ζ2x+ χ2)Arx + (1− rζ2)Ar = 0 implies

that χ2 6= 0 and we can set χ2 = −1 mod G∼; otherwise, we can set χ2 = 0 using P∗(−νχ2),

where ν := −1/ζ2, which preserves Q1. Integrating the system (13) for each of the subcases, werespectively obtain Cases 2 and 3, where ar 6= 0 and thus ar = 1 mod G

∼. In Case 2, one of theconstants aj’s with j < r, a0 or b, if it is nonzero, can be set to ±1 by shifts of x.

k3 = 3. Modulo π∗G∼-equivalence and linearly combining basis elements, we can assume that

Q1 = D(1), Q2 = D(t) + S(ζ2) + P (χ2) and Q3 = D(t2) + S(ζ3) + P (χ3). The completenessof the algebra s with respect to the Lie bracket of vector fields implies that [Q1, Q2] = Q1,[Q1, Q3] = 2Q2 and [Q2, Q3] = Q3. As in the previous case, the first commutation relation holdsonly if ζ2, χ2 = const. The second commutation relation expands to

2D(t) + S(ζ3t ) + P (χ3t ) = 2D(t) + 2S(ζ

2) + 2P (χ2),

and thus ζ3t = 2ζ2, χ3t = 2χ

2. Integrating these two equations yields ζ3 = 2ζ2t + ζ30 andχ3 = 2χ2t+ χ30, where ζ30 and χ30 are constants. Next, commuting Q2 and Q3 yields

[Q2, Q3] = D(t2) + S(2ζ2t) + P (2χ2t+ ζ30χ2 − ζ2χ30).

We thus have [Q2, Q3] = Q3 if and only if ζ30 = 0 and (1+ζ2)χ30 = 0. Considering the classifyingequations (10) for Q1, Q2 and Q3, we obtain the system (13) supplemented by the equationsχ30Ajx = 0, χ30A0x = 4ζ

2 − 2 and χ30Bx + (2ζ2x + 3χ2)A0 = 0. If χ30 6= 0, then ζ2 = −1,

Ajx = 0, and the equations (1 + j)Aj = 0 imply that Aj = 0 for any j, which contradicts thecondition Ar 6= 0. Therefore, χ30 = 0, which requires that ζ2 = 1/2, as well as A0 = 0. We canthen set χ2 = 0 using P∗(2χ

2). The remaining determining equations are Ajx = (j − 2)Aj andBx = −3B, which are readily integrated to Case 4 with ar 6= 0, and hence ar = 1 mod G

∼.

k1 = 0, k2 = 1. Hence Q1 = S(ζ1) + P (χ1), where ζ1 6= 0.

k3 = 0. We can use the adjoint action P∗(χ1/ζ1) to set χ1 = 0. Moreover, multiplying Q1

by a nonzero constant and changing t if ζ1 is not a constant, we can gauge ζ1 to ζ1 = eεt,where ε ∈ {0, 1} mod G∼. The classifying conditions (10) with components of Q1 then implythe system consisting of the equations xAjx = jAj , xA0x = 2ε and xBx − B + εxA

0 = εx. Thegeneral solution of this system is

Aj = αj(t)xj , A0 = α0(t) + 2ε ln |x|, B = β(t)x− ε(α0(t)− 1)x ln |x| − εx ln2 |x|,

where αj , α0 and β are arbitrary smooth functions of t with αr 6= 0. The subgroups of theequivalence group G∼ whose projections to the space (t, x, u) preserve the appropriate subal-gebras s = 〈S(1)〉 and s = 〈S(et)〉 are singled out from G∼ by the constraints X0 = 0 and(X0, Tt) = (0, 1), respectively. Elements from these subgroups allow us to set (α

0, αr) = (0, 1)if ε = 0 or α0 = 1 if ε = 1, which corresponds to Case 5 or 6. In the latter case we have chosenthe gauge α0 = 1 instead of α0 = 0 in order to make the corresponding values of the tuple ofarbitrary elements κ simpler.

k3 = 1. First we reduce, modulo π∗G∼-equivalence, the basis element Q2 with τ2 6= 0 to the form

Q2 = D(1). The Lie bracket [Q2, Q1] = S(ζ1t )+P (χ1t ) is in s provided that the tuples (ζ

1t , χ

1t ) and

20

(ζ1, χ1) are linearly dependent, i.e., ζ1 = c1eεt, χ1 = c0e

εt for some constants c0, c1 and ε, wherec1 6= 0. MultiplyingQ

1 by 1/c1 and, if ε 6= 0, usingD∗(εt), we can set c1 = 1 and ε ∈ {0, 1}. SinceP∗(c0)D(1) = D̃(1) and P∗(c0)Q

1 = S̃(eεt), we can finally reduce Q1 to the form Q1 = S(eεt).Therefore, the corresponding complete system for the arbitrary elements includes the systemfrom the previous case as a subsystem supplemented by the equations Ajt = A

0t = Bt = 0. The

general solution of the complete system is of the same form as in the case k3 = 0, where thefunctions αj , α0 and β should be replaced by the arbitrary constants aj , a0 and b with ar 6= 0.Similarly to the case k3 = 0, the appropriate subalgebras s = 〈S(1),D(1)〉 and s = 〈S(e

t),D(1)〉are preserved by projections of equivalence transformations with (Tt, (X

1t /X

1)t) = (1, 0) and(Tt, (e

−tX1t /X1)t) = (1, 0), which makes possible the gauges (a0, ar) = (0, 1) or a0 = 1 if ε = 0

or ε = 1 and obviously results in Cases 7 and 8, respectively.

k3 > 2. This case cannot be realized. Indeed, otherwise we would additionally have, moduloπ∗G

∼-equivalence, the basis element Q3 = D(t)+S(ζ3)+P (χ3) and would still be able to repeatthe consideration of the case k3 = 1, deriving the expression A

r = arxr with ar 6= 0. At the

same time, the classifying equation (10a) for j = r with the components of Q3 and with theabove Ar reads χ3Arx +A

r = 0, which would imply ar = 0, giving a contradiction.

k1 = 2, k2 = 0. Here we have Qi = P (χi), i = 1, 2. The classifying equations (10) in this case

imply that Ajx = 0, A0x = 0 and χiBx + χ

itA

0 = χitt. Differentiating this last equation withrespect to x leads to Bxx = 0, i.e., B = B

1(t)x+B0(t), and therefore χitt = A0χit +B

1χi.

k3 = 0. We apply S∗(1/χ1) to s and set χ1 = 1, which implies that χ2t 6= 0. Then using the

adjoint action D∗(χ2) we can set χ2 = t. The equations χitt = A

0χit +B1χi, i = 1, 2, then imply

that A0 = B1 = 0. The components of the transformation P(X0) for the arbitrary elements areÃj = Aj , Ã0 = A0 and B̃ = B + X0tt. Thus, choosing X

0tt = −B

0 allows us to gauge B = 0,leading to Case 9.

k3 = 1. We start the simplification of basis elements of s from Q3 with τ3 6= 0, setting, modulo

π∗G∼-equivalence, Q3 = D(1). It then follows from the classifying equations (10) with the

components of Q3 that Ajt = A0t = B

1t = B

0t = 0. We choose

X0 =B0

B1if B1 6= 0; X0 =

B0

A0t if B1 = 0, A0 6= 0; X

0 = −B0

2t2 if B1 = A0 = 0.

Since P∗(X0)D(1) = D̃(1) + P̃ (X0t ), P∗(X

0)P (χi) = P̃ (χi), i = 1, 2, and for the chosen valueof X0 we have X0t ∈ 〈χ

1, χ2〉, the adjoint action P∗(X0) preserves the algebra s. At the same

time, the component of P(X0) for B is B̃ = B +X0tt − A0X0t and therefore B̃

0 = B0 +X0tt −A0X0t −B

1X0 = 0, which yields Case 10, where ar = 1 mod G∼.

k3 > 2. We have one more basis element whose simplified form is Q4 = D(t) + S(ζ4) + P (χ4).

Commuting Q3 and Q4 yields [Q3, Q4] = D(1)+S(ζ4t )+P (χ4t ), which is in the algebra s provided

that ζ4 = const. The commutator of Q4 with Qi, i = 1, 2, gives [Q4, Qi] = P (tχit − ζ4χi), which

is in the algebra s only if χ1 = 1 and χ2 = t up to linearly combining P (χ1) and P (χ2). Itthen follows that A0 = B1 = 0. We can let B0 = 0 mod G∼, and the classifying equation (10c)then implies that χ4tt = 0. Upon linearly combining with P (1) and P (t) we can thus set χ

4 = 0.Moreover, the classifying equation (10a) for j = r with the components of Q4 implies thatζ2 = 1/r and thus, in view of the same equation for other j’s, Aj = 0 if j 6= r. We can gaugeAr = 1 mod G∼. Now suppose that we also have a Q5 = D(t2) + S(ζ5) + P (χ5). Then, theclassifying equations (10) for Ar and A0 require that 2t− rζ5 = 0 and 2ζ5t − 2 = 0. This systemis consistent only for r = 2, where ζ5 = t. This is why Case 11 splits depending on the valueof r. The equation (10c) with A0 = B = 0 implies that χ5tt = 0 and thus we can set χ

5 = 0 uponlinearly combining Q5 with Q1 and Q2.

21

Corollary 20. An rth order evolution equation of the form (1) is reduced to the simplest formut+uux = ur by a point transformation if and only if the dimension of its maximal Lie invariancealgebra is greater than three.

Remark 21. Case 7 can be merged with Case 3 as the particular subcase with ν = 0 if wechoose another second basis element, Q̃2 = −νQ2 = S(1) −D(νt).

Remark 22. Each of the subalgebras of g whose bases are presented in the third column ofTable 1 is really the maximal Lie invariance algebra for the general case of values of the arbitraryelements κ given in the same row. For most of the classification cases, it is not difficult toexplicitly indicate the necessary and sufficient conditions for κ under which there is an additionalLie symmetry extension. They can be found by substituting the corresponding expressions forκ’s components into the system of classifying conditions (10). Thus, these conditions are a0 = 0if ν = −2 or aj = b = 0 with j 6= r if ν = −r with r > 2 in Case 3; r = 2 and b = 0 in Case 4;

αjt = βt = 0 in Cases 5 and 6; aj = 0 for j 6= r and (r−2)2b = (r−1)a20 in Case 10. There are no

additional extensions in Cases 2, 7, 8 and 11. An additional extension exists for Case 9 if and onlyif the parameter functions αj ’s satisfy the system (ζ1t2+τ1t+τ0)αjt+(2ζ

1t+τ1−j(ζ1t+ζ0))αj = 0for some constants ζ0, ζ1, τ0 and τ1. Another form of this condition, which is convenient forchecking, is that

(

αj

αr

)

t

= (r − j)fαj

αr, rαrαjt − jα

jαrt = (r − j)gαjαr,

for some functions f = f(t) and g = g(t) constrained by gtt+3ggt+ g3 = 0, 2(ft+ gf) = gt+ g

2.See also the claim on the most complicated Case 1 in Remark 35.

Remark 23. The proof of Theorem 19 shows that the system of restrictions for appropriatesubalgebras of g〈 〉 presented in Lemmas 15, 17 and 18 is not exhaustive. It can be completedby the conditions restricting the value set of k3 depending on values of k1 and k2,

k3 ∈ {0, 1} if k2 = 1;

if k1 = 2, then k3 ∈ {0, 1, 2} for r > 2 and k3 ∈ {0, 1, 3} for r = 2.

For each value of the tuple k = (k1, k2, k3) satisfying the extended set of restrictions, there existsan appropriate subalgebra of g〈 〉 admitting this value of k.

8 Alternative classification cases

There are various possibilities for choosing representatives in equivalence classes of pairs (s, {Lκ |gκ = s}), where s is an appropriate subalgebra of g. We have tried to simplify the representationof a pair by paying more attention to the pair’s second entry. In most cases the optimal choiceis obvious and coincides with the selection carried out for Table 1. At the same time, there areother options for the proof and representation of results in the cases (k1, k2, k3) = (2, 0, 1) and(k1, k2) = (0, 1).

(k1, k2) = (0, 1). We follow the proof of Theorem 19 but reduce the basis vector fields toanother form. If k3 = 0, we set, modulo G

∼-equivalence, Q1 = S(tε) instead of Q1 = S(eεt),where still ε ∈ {0, 1}. In the case k3 = 1, for Q

2 we choose the form Q2 = D(t) in order to beable to set Q1 = S(tε) again. For ε = 1, this gives the following alternative cases, which arerelated to the corresponding cases of Table 1 by the equivalence transformation with T = eεt,X1 = 1 and X0 = 0:

22

6̃ Aj = αj(t)xj , A0 = α0(t) + 2t−1 ln |x|,

B = β(t)x− α0(t)t−1x ln |x| − xt−2 ln2 |x| S(t)

8̃ Aj = ajt−1xj, A0 = a0t

−1 + 2t−1 ln |x|,

B = bt−2x− a0t−2x ln |x| − xt−2 ln2 |x| S(t),D(t)

(k1, k2, k3) = (2, 0, 1). Instead of setting Q3 = D(1) in the proof for this case, we can simplify

the basis elements Q1 and Q2 to P (1) and P (t) as in the case (k1, k2, k3) = (2, 0, 0). Then readilyAjx = 0, A0 = 0 and, modulo G∼-equivalence, B = 0. Consider the subclass K of equations fromthe class (5) with values of κ satisfying the above constraints,

K = {Lκ | Ajx = 0, A

0 = 0, B = 0, Ar 6= 0}.

The subclass K turns out to be normalized with respect to its usual equivalence group G∼K, whichis finite-dimensional and consists of the transformations in the space of (t, x, u,A2, . . . , Ar) whosecomponents are of the form (6a)–(6b) with

T =c1t+ c2c3t+ c4

, X1 =c5

c3t+ c4, X0 =

c6t+ c7c3t+ c4

,

where c1, . . . , c7 are arbitrary constants with (c1c4−c2c3)c5 6= 0 that are defined up to a nonzeromultiplier. Hence the equivalence algebra g∼K of K is spanned by D̂(1), D̂(t), D̂(t

2)+ Ŝ(t), Ŝ(1),

P̂ (1) and P̂ (t). The kernel Lie invariance algebra of K is g∩K = 〈P (1), P (t)〉. The restrictionof G∼-equivalence to the subclass K coincides with G∼K-equivalence. This is why, up to G

∼-equivalence, we can assume that in this case the appropriate subalgebra s is spanned by Q1 =P (1), Q2 = P (t) and Q3 = D(τ3)+S(ζ3), and there are three cases for (τ3, ζ3) and (A2, . . . , Ar):

(a) τ3 = 1, ζ3 = σ ∈ {0, 1}, Aj = ajejσt;

(b) τ3 = t, ζ3 = σ = const > 0, Aj = ajt−1|t|jσ;

(c) τ3 = t2 + 1, ζ3 = σ = const > 0, Aj = aj(t2 + 1)j/2−1ejσ arctan t.

Therefore, instead of the single Case 10 we have the following three cases:

˜10a Aj = ajejσt, A0 = B = 0 P (1), P (t),D(1) + S(σ)

˜10b Aj = ajt−1|t|jσ, A0 = B = 0 P (1), P (t),D(t) + S(σ)

˜10c Aj = aj(t2 + 1)j/2−1ejσ arctan t, A0 = B = 0 P (1), P (t),D(t2 + 1) + S(t+ σ)

Modulo G∼-equivalence, σ ∈ {0, 1}, σ > 0 and σ > 0 in Cases ˜10a, ˜10b and ˜10c, respectively.

The advantage of this form for Case 10 is that the vector fields Q1 and Q2 then have theevident interpretation as generators of translations with respect to the space variable x andGalilean boosts, respectively. Meanwhile the forms of Q3 and, especially, Aj become morecomplicated. This is why we chose the previous form of Case 10 for Table 1. Subcases of thiscase are related to subcases of the alternative Case 1̃0 by equivalence transformations of theform (6), where X0 = 0 and

10a → ˜10b: T = e(λ2−λ1)t, X1 = e−λ1t, σ = −λ1/(λ2 − λ1);

10b → ˜10a: T = t, X1 = e−µt, σ = −µ;

10c → ˜10c: T = tan νt, X1 = e−µt/ cos νt, σ = −µ/ν.

23

9 Equations with time-dependent coefficients

To study Lie symmetries of equations from the class (1) with coefficients depending at most on t,it is again convenient to start with a wider class, which is the subclass K0 of the class (1) singledout by the constraint Cx = 0 (resp. A

rx = 0) implying Xxx = 0 for admissible transformations.

Proposition 24. The class K0 is normalized in the usual sense. Its usual equivalence group isconstituted by the transformations of the form

t̃ = T (t), x̃ = X1(t)x+X0(t), ũ = U1(t)u+ U0(t, x),

Ãj =(X1)j

TtAj , Ã1 =

X1

Tt

(

A1 +U0

U1C −

X1t x+X0t

X1

)

, Ã0 =1

Tt

(

A0 +U1tU1

+U0xU1

C

)

,

B̃ =U1

TtB +

U0tTt

+U0xTt

(

U0

U1C −

X1t x+X0t

X1

)

−r∑

k=0

U0k(X1)k

Ãk, C̃ =X1

TtU1C,

where j = 2, . . . , r, and T = T (t), X1 = X1(t), X0 = X0(t), U1 = U1(t) and U0 = U0(t, x) arearbitrary smooth functions of their arguments with TtX

1U1 6= 0.

Consider the subclass K1 obtained by attaching the constraints A0x = 0, A

1xx = 0, A

jx = 0,

j = 2, . . . , r, Cx = 0 and Bxx = 0 to the auxiliary system for arbitrary elements. It is alsonormalized in the usual sense and its usual equivalence group is the subgroup of the usualequivalence group G∼K0 of the class K0 that is associated with the constraint U

0xx = 0, i.e.,

U0 = U01(t)x + U00(t). Note that we can reparameterize the class K1 by representing B =B1(t)x + B0(t), A1 = A11(t)x + A10(t) and assuming the coefficients B1, B0, A11 and A10 asarbitrary elements instead of B and A1. The transformation component for B simplifies to

B̃ =U1

TtB +

U0tTt

−U0xTtA1 −

U0

Tt

(

A0 +U1tU1

+U0xU1

C

)

.

The next intermediate subclass K2 is singled out by strengthening the constraint for A1 to

A1x = 0. In fact, this can be realized by gauging A1 in the class